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Boolean Games

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Boolean Games

1. 1. 1vs. Boolean Games turn based, one on one
2. 2. Exercise 51
3. 3. Players take turns Exercise 51
4. 4. Players take turns 4 connected win Exercise 51
5. 5. Players take turns 4 connected win Exercise 51
6. 6. have Fun Players take turns 4 connected win Exercise 51
7. 7. 1 0 1 0 1 0
8. 8. 1 0 1 0 1 0 0 starts
9. 9. 1 0 1 0 1 0 1 0 1 0 1 0 00 starts
10. 10. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 starts 0 starts
11. 11. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 11 starts 0 starts
12. 12. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 101 starts 0 starts
13. 13. Task: construct a set of Horn clauses that describe if a player has already won or lost.
14. 14. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 101 starts 0 starts
15. 15. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 101 starts 0 starts 0
16. 16. 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 101 starts 0 starts 0 0
17. 17. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 1 starts 0
18. 18. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 1 starts 0
19. 19. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 1 starts 0
20. 20. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 1 starts 0
21. 21. How can we generalize?
22. 22. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 0
23. 23. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 0
24. 24. 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 10 0 if a draw is not possible
25. 25. 00001110001000110000111000001100011100001110000001 10001000011000110000110000011000001000110000001000 00111000000111000100011000011100000110001110000111 00000011000100001100011000011000001100000100011000 00010000011100000011100010001100001110000011000111 00001110001000110001000011000110000110000011000001 00011000000100000111000000111000100011000011100000 11000111000011100000011000100001100011000011000001 10000010001100000010000011100000011100010001100001 11000001100011100001110000001100010000110001100001 10000011000001000110000001000001110000001110001000 11000011100000110001110000111000000110001000011000 11000011001000110000010001100002001000001110000001 11000100011000011100000110001110000111000000110001 00001100011000011000001100000100011000000100000111 00000011100010001100001110000011000111000011100000 “Any Boolean function leads to a Game ...” Donald E. Knuth Exercise 52
26. 26. construct game graph animate x2 start
27. 27. construct game graph animate x2 x1 start
28. 28. construct game graph animate x2 x1 x4 start
29. 29. construct game graph animate x2 x3 x5 x7 x6 x8 x9x1 x4 start
30. 30. x1 x3 x5 x7 x6x4 x8 x9 1 some example steps - max 4 possible moves - start
31. 31. x1 x3 x5 x7 x6 x8 x9 1 0 some example steps - max 4 possible moves - start
32. 32. x1 x5 x7 x6 x8 x9 1 0 1 some example steps - max 4 possible moves - start
33. 33. x1 x5 x7 x6 x8 x9 0 1 0 some example steps - max 4 possible moves - start
34. 34. x1 x2
35. 35. x1 x2 (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
36. 36. x1 x2 xx (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
37. 37. x1 x2 xx x0 x1 (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
38. 38. x1 x2 xx x0 x1 00 10 01 11 (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
39. 39. x1 x2 xx x0 x1 0 0 0 1 (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
40. 40. x1 x2 xx x1 0 0 0 1 0 (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
41. 41. Algorithm C
42. 42. n (a) 2 0 wins 3 0 wins 4 ﬁrst wins 5 second wins 6 second wins 7 1 loses if ﬁrst 8 draw 9 draw (a) f(x[1:n]) = x[1:n] < x[n:1]╓ ╙ ╓ ╙
43. 43. n (b) 2 second wins 3 ﬁrst wins 4 ﬁrst wins 5 draw 6 second wins 7 second wins 8 draw 9 draw (b) f(x[1:n]) = xi⊕i
44. 44. n (c) 2 1 wins 3 ﬁrst wins 4 ﬁrst wins 5 draw 6 1 loses if ﬁrst 7 1 loses if ﬁrst 8 draw 9 draw (c) f(x[1:n]) = x[1:n] contains no two consecutive 1’s ╓ ╙ ╓ ╙
45. 45. n (d) 2 second wins 3 ﬁrst wins 4 ﬁrst wins 5 1 loses if ﬁrst 6 1 loses if ﬁrst 7 1 loses if ﬁrst 8 1 loses if ﬁrst 9 1 loses if ﬁrst (d) f(x[1:n]) = (x[1:n])2 is prime╓ ╙ ╓ ╙
46. 46. Questions?